Travelling Waves Solution of the Unsteady Problem of Binary Gas Mixture Affected by a Nonlinear Thermal Radiation Field
In the present study, a development of the paper [Can. J. of Phy., 2012, 90(2): 137-149] is introduced. The non-stationary BGK (Bhatnager- Gross- Krook) model of the Boltzmann nonlinear partial differential equations for a rarefied gas mixture affected by nonlinear thermal radiation field, for the first time, are solved instead of the stationary equations. The travelling wave solution method is used to get the exact solution of the nonlinear partial differential equations. These equations were produced from applying the moment method to the unsteady Boltzmann equation. Now, nonlinear partial differential equations should be solved in place of nonlinear ordinary differential equations, which represent an arduous task. The unsteady solution gives the problem a great generality and more applications. The new problem is investigated to follow the behavior of the macroscopic properties of the gas mixture such as the temperature and concentration. They are substituted into the corresponding two stream Maxiwallian distribution functions permitting us to investigate the non-equilibrium thermodynamic properties of the system (gas mixture + the heated plate). The entropy, entropy flux, entropy production, thermodynamic forces, kinetic coefficients are obtained for the mixture. The verification of the Boltzmann H-theorem, Le Chatelier principle, the second law of thermodynamic and the celebrated Onsager’s reciprocity relation for the system, are investigated. The ratios between the different contributions of the internal energy changes based upon the total derivatives of the extensive parameters are estimated via the Gibbs formula. The results are applied to the Argon-Neon binary gas mixture, for various values of both of the molar fraction parameters and radiation field intensity. Graphics illustrating the calculated variables are drawn to predict their behavior and the results are discussed.
Taha Zakaraia Abdel Wahid,
Travelling Waves Solution of the Unsteady Problem of Binary Gas Mixture Affected by a Nonlinear Thermal Radiation Field, American Journal of Physics and Applications.
Vol. 2, No. 6,
2014, pp. 121-134.
A. Beskok, G.E. Karniadakis, W. Trimmer, Rarefaction and compressibility effects in gas micro-flows, Trans. ASME 118 (3) (1996) 448–456.
C.M. Ho, Y.C. Tai, Micro-electro-mechanical systems (MEMS) and fluid flows, Ann. Rev. Fluid. Mech. 30 (1998) 579–612.
S. Naris, D. Valougeorgis, D. Kalempa, and F. Sharipov "Gaseous mixture flow between two parallel plates in the whole range of the gas rarefaction "Physica A, 336, (2004) 294-318.
Taha Zakaraia Abdel Wahid" Travelling Waves Solution of the Unsteady Flow Problem of a Rarefied Non-Homogeneous Charged Gas Bounded by an Oscillating Plate." Mathematical Problems in Engineering 2013; 2013(ID 503729):1-13.
Taha Zakaraia Abdel Wahid" Kinetic and Irreversible Thermodynamic study of Plasma and Neutral Gases." LAMBERT Academic Publishing, Germany, (2014).
F. Sharipov, D. Kalempa, Velocity slip and temperature jump coefficients for gaseous mixtures. III. Diffusion slip coefficient, Phys. Fluids 16 (10), (2004), 3779-3785.
F.J. McCormack, Construction of linearized kinetic models for gaseous mixture and molecular gases, Phys. Fluids 16 (1973) 2095–2105.
S. Takata, Diffusion slip for a binary mixture of hard-sphere molecular gases: numerical analysis based on the linearized Boltzmann equation, in: T.J. Bartel, M.A. Gallis (Eds.), Rarefied Gas Dynamics, Vol. 585, 22nd International Symposium, AIP Conference Proceedings, New York, 2001, pp. 22–29.
S. Takata, S. Yasuda, S. Kosuge, K. Aoki, Numerical analysis of thermal-slip and diffusion-slip flows of a binary mixture of hard-sphere molecular gases, Phys. Fluids A 15 (2003) 3745–3766.
V.G. Chernyak, V.V. Kalinin, P.E. Suetin, The kinetic phenomena in non-isothermal motion of a binary gas mixture through a plane channel, Int. J. Heat Mass Transfer 27 (8) (1984) 1189–1196.
D. Valougeorgis, Couette _ow of a binary gas mixture, Phys. Fluids 31 (3) (1988) 521–524.
F. Sharipov, D. Kalempa, Gaseous mixture flow through a long tube at arbitrary Knudsen number, J. Vac. Sci. Technol. A 20 (3) (2002) 814–822.
V. Garz, A. Santos, J.J. Brey, A kinetic model for a multicomponent gas, Phys. Fluids A 1 (2) (1989), 380–383.
A.M. Abourabia and T. Z. Abdel Wahid" The Unsteady Boltzmann Kinetic Equation and Non-Equilibrium Thermodynamics of an Electron Gas for the Rayleigh Flow Problem", Can. J. Phys. 88: (2010), 501–511.
A. M. Abourabia and T. Z. Abdel Wahid "Kinetic and Thermodynamic Treatment for The Rayleigh Flow Problem of an Inhomogeneous Charged Gas Mixture", J. Non-Equilib. Thermodyn. 37 (2012), 119–141.
S.R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics, Dover Inc., New York, 1984.
C. Cercignani, The Boltzmann Equation and its Application, Springer, New York, 1988.
G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford, 1994.
Taha Zakaraia Abdel Wahid "Travelling Wave Solution of The Unsteady BGK Model for a Rarefied Gas Affected by a Thermal Radiation Field.", (Accepted) Sohag Journal of Mathematics, (2014).
T. Z. Abdel Wahid and S.K. Elagan "Exact Solution of the Unsteady Rayleigh Flow Problem of a Rarefied charged Gas bounded by an oscillating plate." Can. J. Phys. 90: 987–998 (2012).
A. M. Abourabia and T. Z. Abdel Wahid "Solution of The Krook Kinetic Equation Model and Non-Equilibrium Thermodynamics of a Rarefied Gas Affected by a Nonlinear Thermal Radiation Field.", J. Non-Equilib. Thermodyn. 36 (2011), 75–98.
A.M. Abourabia and T. Z. Abdel Wahid "The unsteady Boltzmann kinetic Equation and non-equilibrium thermodynamics of an electron gas for the Rayleigh flow problem "Can. J. Phys. vol 88, (2010), 501–511.
T. Z. Abdel Wahid "Thermodynamic Treatment for the Exact Solution of the Unsteady Rayleigh Flow Problem of a Rarefied charged Gas." J. Non-Equilib. Thermodyn. 37 (2012), 119–141.
V. Garzo and A. Santos " Heat and momentum transport in a gaseous dilute solution" Phys. Rev. E, 48(1), (1993) , 256-262.
F. G. Cheremisin, "Verification Of The Nature Of The Approximation Of The Boltzmann Integral By Krook's Kinetic Relaxation Model" Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, (4) (1970), 3-7.
F. Sharipov and G. Bertoldo, Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number. Phys. Fluids 21, 067101, (2009).
F. Sharipov and G. Bertoldo," Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential." J. Comp. Phys.228(9), (2009), 3345-3357.
S. Chapman and T. G. Cowling, "Mathematical Theory of Non-Uniform Gases", Cambridge University Press, Cambridge, (1970).
F. Sharipov, L. M. G. Cumin and D. Kalempa," Heat flux through a binary gaseous mixture over the whole range of the Knudsen number." Physica A 378 , (2007), 183-193.
C. Marin, V. Garzo, and A. Santos “Nonlinear Transport in a Dilute Binary Mixture of Mechanically Different Particles” J. Stat. Phys., 75(5/6) (1994), 797-816.
P. L. Bhatnagar, E. P. Gross and M. Krook, "A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems" Phys. Rev., 94 (1954), 511-525.
E. M. Shakhov "Generalization of The Krook Kinetic Relaxation Equation" Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, 3(5), 1968, 142-145.
C. Cercignani, "Slow Rarefied Flows: Theory and Application to Micro-Electro- Mechanical Systems " Birkhauser - Verlag, Basel ,(2006).
T. Z. Abdel Wahid "Exact Solution of The Unsteady Krook Kinetic Model and Non-Equilibrium Thermodynamic Study for a Rarefied Gas Affected by a Nonlinear Thermal Radiation Field. ", Canadian Journal of Physics (2013); 91(3):201-210.
Sebastian Volz (Ed.) " Microscale and Nanoscale Heat Transfer" Springer-Verlag Berlin Heidelberg ,(2007).
V. P. Shidloveskiy, “Introduction to Dynamics of Rarefied Gases”, Elsevier NY, USA, (1967).
S. P. Dawson and A. D. Verga," On The Theory Of Helium Diffusion In Stellar Outer Layers "Rev. Mexicana Astron. Astrof.,13(1986),.Provided by NASA Astrophysics Data System , 85-100.
Doctor of Philosophy Thesis by Mitchell Thomas," Radiation Transfer and Opacity Calculations", California Institute of Technology , Pasadena, California. Page 26,(1964).
D. Mihalas and B.W. Mihalas," Foundation of radiation hydrodynamics " Oxford , NY.,UN. Press, (1984).
M. M. Elafify, K. M. Elawadly and Sh.G. Elsharkawy" Prediction of the thermal conductivity of binary gas mixtures using direct simulation Monte Carlo method" Info. Tech. Jou., 3(1), (2003), 23-35.
J. M. Burt and I. D. Boyd "Evaluation of particle method for the ellipsoidal statistical Bhatnagar Gross Krook equation" the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV. , AIAA 2006-989.
Lees, L. and Liu, C.-Y., “Kinetic Theory Description Of Conductive Heat Transfer From A Fine Wire”, Physics of Fluids, 5(10), (1962), 1137-1148.
Lees, L. “Kinetic Theory Description of Rarified Binary gas mixturees” Journal of The Society for Industrial and Applied Mathematics, 13(1), (1965), 278-311.
J.C. Maxwell, L. Rayleigh "Theory of Heat" Longmans, Green, AND Co., London, UK. (1902).
E.M. Shakhov," Couette Problem for The Generalized Krook Equation Stress-Peak Effect "Izv. AN SSR. Mekhanika Zhidkost I Gaza, 4,(5), (1969), 16-24.
Jeans, J. "The Dynamical Theory of Gases." Cambridge University Press, Cambridge, (1904).
V. V. Aristov "Methods of Direct Solving The Boltzmann Equation and Study of Non-equilibrium", Kluwer Academic, Dordrech, Netherland, 2001.
V. Garzo and A. Santos " Heat and momentum transport in a gaseous dilute solution" Phys. Rev. E, 48(1), (1993) , 256-262.
A. Santos, J. J. Brey “Kinetic model for steady heat flow" Phys. Rev. E, 34(6) (1986), 5047-5050.
C. Marin and V. Garzo "Exact solution of the Gross–Krook kinetic model for a multicomponent gas in steady Couette flow”, Physica A 312, (2002), 315 – 341.
Gratton, J., Mahajan, S. M. and Minotti, F., "Non-Newtonian Gravity Creeping Flow", pp. 1–17, International Centre for Theoretical Physics, Trieste, 1988.
Nugroho, G., Ali, A. M. S. and Abdul Karim, Z. A., "Towards a new simple analytical formulation of Navier–Stokes equations", World Acd. Sci., Eng. Tec., 51 (2009), 190–194.
Clause, P. J. and Mareschal, M., Heat transfer in a gas between parallel plates: Moment method and molecular dynamics, Phys. Rev. A, 38(8) (1988), 4241–4252.
Ying-Chu L. W., Flow Generated by a Suddenly Heated Flat Plate, PhD thesis, California Institute of Technology, Pasadena, CA, 1963.
B. Chan Eu, "Kinetic Theory and Irreversible Thermodynamics", Wiley, NY, USA, 1992.
D. Jou and J. Casas-Vázquez, G. Lebon, "Extended Irreversible Thermodynamic", Springer -Verlag, Berlin, Germany, (1993).
V. Zhdanov and V. Roldughin , "Non-Equilibrium Thermodynamics And Kinetic Theory of Rarefied Gases" ,Physics Uspekhi, 4 (4) (1998), 349-378.
V. M. Zhdanov and V. I. Roldugin, “Nonequilibrium Thermodynamics of a Weakly Rarefied Gas Mixture” Zh. Eksp. Teor. Fiz., 109, (1996), 1267–1287.
G. Lebon, D. Jou, J. Casas-Vàzquez "Understanding Non-equilibrium Thermodynamics : Foundations, Applications, Frontiers "Springer-Verlag Berlin ,Heidelberg, (2008).
F. Sharipov "Reciprocal relations based on the non-stationary Boltzmann equation", Physica A 391, 1972-1983 (2012).
R. M. Velasco, L. S. García-Colín and F. J. Uribe " Entropy Production: Its Role in Non-Equilibrium Thermodynamics" Entropy 2011, 13, 82-116; (doi:10.3390/e13010082)
A. B. De Castro, "Continuum Thermomechanics" Birkhauser, Verlag, Basel, Switzerland, (2005).
G. M. Alves and G. M. Kremer," Classical Kinetic Theory for Binary Gas Mixtures of Monatomic and Polyatomic Gases.", J. Non-Equilib. Thermodyn.,17(2),(1992), 171-182.
C. F. Delale "The Generalized H Theorem in the Hilbert Space", J. Chem. Phys., 84 (2), (1986), 971-975. (doi:10.1063/1.450545)
S. Sieniutycz and R. S. Berry "Canonical formalism, fundamental equation, and generalized thermomechanics for irreversible fluids with heat transfer"PHY. REV. VOL. 47, N. 3 , 1993.
T.D. Frank "Pumping and Entropy Production in Non-equilibrium Drift-Diffusion Systems: A Canonical-Dissipative Approach" Eur. J. of Sci. Res., Vol.46 No.1 (2010), 136-146.
L. M. Martyushev, V. D. Seleznev " Maximum entropy production principle in physics, chemistry and biology " Physics Reports, 426, (2006), 1 – 45.
V.I. Roldughin, V.M. Zhdanov"" Non-equilibrium thermodynamics and kinetic theory of gas mixtures in the presence of interfaces" Advances in Colloid and Interface Science, 98,(2002),121-215.
Sharipov, F., Onsager–Casimir reciprocity relations for a mixture of rarefied gases interacting with a laser radiation, J. Stat. Phys., 78(1/2) (1995), 413–430.
Sharipov, F. M., Onsager–Casimir reciprocity relations for open gaseous system at arbitrary rarefaction, Phys. A, 203(1994), 437–456.
W. W. Liou and Y. Fang, “Microfluid Mechanics principles and Modeling" The McGraw-Hill Companies, Inc, NY, USA, 2006.
S. Narisa, D. Valougeorgisa; D. Kalempa and F. Sharipov "Gaseous mixture flow between two parallel plates in the whole range of the gas rarefaction "Physica A, 336, (2004), 294 – 318.
J.D. Huba., NRL plasma formulary., Naval Research Laboratory, Washington, D.C. 2011.